A380979 Composites that cause a witness to be added to a set of Fermat witnesses: a(n) is the smallest composite number that is not guaranteed composite using Fermat's Little Theorem by the witness A380978(i) for any i < n.

4, 341, 1105, 1729, 29341, 75361, 162401, 252601, 294409, 334153, 399001, 1152271, 1615681, 2508013, 3581761, 3828001, 6189121, 6733693, 10024561, 10267951, 14469841, 17098369, 17236801, 19384289, 23382529, 29111881, 34657141, 53711113, 64377991, 79411201, 79624621

Offset: 1

Author

Jan Kostanjevec, Feb 10 2025

Keywords

• nonn

Comments

A380978(n) is defined as the minimal Fermat witness that guarantees the compositeness of a(n). See the Weisstein link for details of the guarantee -- the option that uses a property derived from Fermat's little theorem.

To what extent does this differ from A135720 sorted? - Peter Munn, Mar 12 2025

Examples

			a(1) = 4, since 4 is the smallest composite number and we need to add a witness to the empty set to guarantee its compositeness. 2 is the minimal Fermat witness for the compositeness of 4, so the set of witnesses becomes {2}.
a(2) = 341, since 341 is the smallest composite number that requires a witness other than 2, namely 3.
a(3) = 1105, since 1105 is the smallest composite number that requires a witness other than 2 and 3, namely 5.
		

Links

• Eric Weisstein's World of Mathematics, Compositeness Certificate
• Index entries for sequences related to pseudoprimes

Crossrefs

Cf. A001567, A002997, A006945, A098654, A135720, A380978 (new minimal Fermat witness).

Extensions

More terms from Jinyuan Wang, Mar 05 2025